2 1. MATHEMATICAL SHAPES OF UNCERTAINTY Following the ideas of Wiener, the original mathematical version of the Uncer- tainty Principle can be formulated as follows: A function (measure, distribution) and its Fourier transform cannot be simultaneously small. The informal term ‘small’ in this statement can be given a variety of mathematical meanings, each leading to a deep and important problem of analysis, see for instance [60]. Examples of such problems will be discussed in section 2. As it often happens in mathematics, these problems can then be viewed in more general and abstract settings, sometimes developing into a whole branch of analysis. One of such branches is the study of ‘basis’ properties of more and more gen- eral families of functions. Via its very definition, the Fourier transform is directly related to the family of complex exponentials and questions on bases, frames and completeness of families of such functions (some of them will be discussed below). But the exponentials are not the only natural family for such studies and similar questions can be asked about every new set of ‘harmonics.’ Following this direction one may arrive at other classical problems, such as the moment problem, prob- lems of completeness of various families of polynomials and special functions, and proceed further to wavelet expansions or other bases and frames, see for instance [24, 39]. Another example of broader meaning of the Uncertainty Principle is the clas- sical theory of zero sets of entire functions. If a system of exponentials E = {eiλnt} is incomplete in L2([−1, 1]) then this space must contain a non-trivial function f orthogonal to all the exponentials in the family. Then the Fourier transform ˆ of this function is an entire function that vanishes at {λn}. To establish completeness of E in L2([−1, 1]), one needs to prove a version of the Uncertainty Principle: f and ˆ cannot be simultaneously small, in the sense that f cannot have a small support, contained in the interval [−1, 1], while ˆ is being small by vanishing on the large set {λn}. Problems of this sort were studied in the 1930’s and 40’s by Paley and Wiener, Levinson and many other prominent mathematicians, see for instance [97]. The smallness of the support of f immediately translates into the exponential type of the entire function ˆ, i.e. into a restriction on its growth in the complex plane. After that one can use use standard analytic tools, such as Jensen’s inequalty, to obtain restrictions on density and balance of the zero set {λn}. More advanced methods give further, more subtle results, while the problem gradually expands be- yond the Paley-Wiener classes, i.e. beyond Fourier transforms of functions from L2 on an interval. A more general question is how the growth estimates translate into restrictions on the zero sets for broader classes of entire functions. Even though the Fourier transform disappears from this general formulation, it is clearly still a version of the Uncertainty Principle. Such problems were at the center of complex analysis for a large part of the 20th century, e.g. [94]. Further generalizations of the Uncertainty Principle (UP) include Fourier anal- ysis on groups (see [49, 50]), problems in signal processing and time-frequency anal- ysis stemming from a seminal paper by Gabor [51] (see also [11]), and many other important problems of analysis. At present, a meaningful survey of all branches of UP would require several volumes. In these notes we certainly do not aim at giving such a survey. Fortunately, many good texts on various topics of UP are available. Let us mention the book by Havin and J¨ oricke [60] and the paper by Folland and Sitaram [50] (which, to a certain extent, complement each other) as excellent initial

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